Seismic waves generated artificially for the imaging of geological layers have been used for more than 50 years. During seismic explorations, as shown in FIG. 1, a controlled source 2 generates seismic waves 4, which propagate through the earth. Some of these seismic waves are reflected at the interface(s) 6 between different geological layers, e.g., geological layer 8 and geological layer 10. As the reflected waves 12 return to the surface of the ground 16, the waves are detected by receivers 14 which convert the seismic energy into an electrical signal which is then recorded by recording equipment. Analysis of travel times and amplitudes of these waves makes it possible to construct a representation of the geological layers on which the waves reflect.
When a seismic survey is conducted on land, the seismic source is usually a controlled explosion, a vibrator, or a weight-drop. When the seismic survey is conducted in a marine environment, the seismic source is typically an air gun. For the explosive source type, dynamite is buried in a hole several meters deep. The hole is filled in and then the dynamite is detonated. Vibrator sources work by shaking the ground for several seconds, at a variety of frequencies. Weight-drop sources involve dropping a large weight onto the ground. Air guns are positioned beneath the surface of the water, where they release a burst of compressed air to generate acoustic waves. Other source types, such as, using the earth's natural rumblings as a passive source, can also be used. However, land seismic surveys predominantly use either explosive or vibrator sources, while marine surveys predominantly use air guns.
For land seismic surveys, geophones are used as receivers. Geophones convert the ground displacement (caused by the elastic seismic waves) into a voltage. For marine surveys, hydrophones, which are also used as receivers, convert pressure changes (such as those caused by underwater sound waves) into an electrical signal. For both types of receivers, the variation of the received signal from a baseline is recorded digitally. This recording from one receiver is called a seismic trace. A seismic trace generally extends over several seconds of listening time.
Seismic waves that are reflected once from an interface are called primary reflections and are of value when trying to construct a representation of the geological layers. An example of this is shown in FIG. 2 wherein if the interface 6 is smooth and continuous at the point of reflection 18, the angle 22 at which the wave reflects is equal to its angle of incidence 20. These so-called “specular reflections” are used to construct the coarse-scale features of the layers, i.e., the general structure and global trends of the layers. If the point of reflection 24 is a discontinuity, a diffraction results, in which the incident energy is scattered in all directions. Diffractions indicate the presence of fine-scale features such as faults, fractures and abrupt edges.
Numerous other waves arrive at and are recorded by the receiver(s) 14. For example, air blast noise refers to waves generated by the source which travel through the air to the receiver 14. Ground roll waves are source-generated waves that propagate along the surface of the ground. Multiples are waves that are reflected more than once. Background noise encompasses non-source-generated waves, such as, from wind, rain, ocean waves, power lines, traffic, humans, animals as well as noise generated by the recording instruments and receivers. A goal of seismic processing is to extract the useful energy from the seismic traces and to infer from that energy, a picture of the layers beneath the sources and receivers.
Many seismic processing algorithms make the simplifying assumption that the waves arriving at the receiver are plane waves. A plane wave is a theoretical construct that has a wavefront with no curvature, i.e., the wavefront is planar. This assumption is approximately realistic for seismic data because the distances between source and receiver are typically large distances. The actual wavefront has a large enough radius that the portion of the wavefront that reaches the receiver is very near planar.
To differentiate signal from noise, seismic experiments employ a high degree of redundancy. That is, a seismic survey attempts to sample each reflection point many times. To accomplish this, seismic waves generated by a single source point are recorded at many receiver locations, and the experiment is repeated for many source locations. As shown in FIG. 3, current land three dimensional (3D) seismic methods lay out receivers over the survey area in an organized grid made up of receiver lines, while source points are laid out on an overlapping grid make up of shot lines. The term offset is used herein to refer to the distance between a source and a receiver. Azimuth is the angular direction, in degrees measured from north, between a source and receiver. The “inline” direction is parallel to the receiver lines, while the “crossline” direction is perpendicular to the receiver lines. FIG. 3 shows an orthogonal survey 26 in which the shot lines 28 are orthogonal to the receiver lines 30. In FIG. 3, the receiver lines 30 have black triangles representing the receivers and the shot lines 28 have circled asterisks representing the shots. A common midpoint (CMP) grid 32 overlays the orthogonal survey 26.
In seismic surveys, the source and receiver locations are known but the point of reflection is not known. Seismic processing methods initially assume that the reflection point occurs midway between the source and receiver. The survey area is divided into a grid of CMP bins, e.g., bins 34 and 36, as shown in FIG. 3, and the seismic traces are each assigned to the bin containing the seismic trace's midpoint. Seismic traces that fall into the same CMP bin form a CMP gather. The number of traces in a gather is called the gather's fold. Traces in one CMP gather can be further organized by their respective offsets and azimuths. Alternatively, the traces in one CMP gather can be organized into “common offset vectors” (COVs), by replacing offset and azimuth with a 2-component offset vector. The first component is the offset in the inline direction and the second component is the offset in the crossline direction.
Redundancy provides an opportunity to examine how amplitudes in a CMP gather vary with changing offset and azimuth. In general, longer offsets imply smaller angles of incidence. The amount of reflected energy at an interface changes with angle of incidence and is measured by changes in amplitudes of the seismic traces in a CMP gather. An amplitude-versus-offset (AVO) analysis can infer properties of the interface, including rock density, porosity and fluid content. Amplitude-versus-azimuth (AVAz) analysis infers fracture orientation and other heterogeneities within layers. To provide an image of the subsurface, however, the redundancy is removed with a “stacking” process, which averages all traces in each gather after correcting for their different travel paths. Stacking is a powerful tool for removing noise, while reinforcing reflected signal(s).
The initial assumption that reflection points lie midway between the source and receiver is not true in general. When a geological layer 38 is perfectly flat and the layer(s) above are homogeneous, as shown in FIG. 4, the reflection point 42 is beneath the midpoint 40. Layer 46, which is dipping with respect to the surface 45 as shown in FIG. 5, tends to have reflection points that are not beneath the midpoints. Geological layers which are non-homogeneous, in particularly those for which the variations cause waves to be transmitted at different velocities, may also give reflection point(s) 44 at location(s) other than the midpoint 48. Processing of seismic data usually includes a seismic migration step in which a migration algorithm moves the energy from the common midpoint to the actual reflection point 44. Migration algorithms also collapse diffractions to their scattering point. Migration algorithms are categorized by where they are performed in the processing sequence relative to the stacking step. A pre-stack time migration (PSTM) performs migration on the CMP gathers prior to stacking.
For optimal imaging, PSTM algorithms generally require a uniform sampling of offsets and azimuths for each CMP. However, orthogonal land seismic surveys often produce data that is sparsely and irregularly sampled in the offset and azimuth dimension due to high cost and access restrictions in the field. One method to ameliorate this issue is to use a so-called “5 dimensional (D)” interpolation, which mathematically reconstructs the missing traces to provide the regular and uniform spatial sampling in four spatial dimensions that benefits pre-stack migration. Five dimensional (5D) interpolation can also provide better conditions for AVO and AVAz analysis. One problem associated with 5D interpolation in general is that fine-scale details, e.g., diffractions and noise, may not interpolate well. While these details are not lost on the original traces, their energy becomes diluted when migrated with the interpolated traces.
There are several 5D interpolation algorithms in commercial use, including Minimum Weighted Norm Fourier Interpolation (MWNI) (LIU, B and SACCHI, M., 2004, “Minimum weighted norm interpolation of seismic records,” Geophysics, 69, pp. 1560-1568; TRAD, D., 2008, “Five-dimensional seismic data interpolation,” 78th Annual International Meeting,” SEG, Expanded Abstracts, pp. 978-982), Anti-Leakage Fourier Transform (ALFT) (XU, S., ZHANG, Y., and LAMBARE, G., 2004, “On the orthogonality of anti-leakage Fournier transform based seismic trace interpolation,” 74th Ann. International Meeting” SEG, Expanded Abstracts (SP 3.7; XU, S., ZHANG, Y., and LAMBARE, G., “2010, “Antileakage Fournier transform for seismic data regularization in higher dimensions,” Geophysics 75, WB113-120) and Projection onto Convex Sets (POCS) (ABMA, R. and KABIR, N., 2006, “3D interpolation of irregular data with a POCS algorithm,” Geophysics 71, E91-E97). All these methods rely on the idea that the wavefield to be reconstructed is a superposition of plane waves. Cadzow interpolation (TRICKETT, S., BURROUGHS, L., and MILTON, M., 2010, “Rank-reduction-based trace interpolation,” 80th Annual International Meeting, SEG, Expanded Abstracts, pp. 3829-3833) is potentially more flexible. Nevertheless, Cadzow interpolation's well understood properties also assume a plane-wave model.
Interpolation methods based on plane waves are good at reconstructing smooth global trends, however they are not efficient at reconstructing the fine-scale features that provide spatial resolution. This can be demonstrated by comparing a 3D model which includes two coarse-scale events and one fine-scale feature. A cross section from the 3D model is shown in FIGS. 6-9. FIG. 6 shows a model 50 with three events, two coarse-scale events 52, 54 and one fine-scale event 56. FIG. 7 shows a cross-section 58 from a 3D volume in which the volume is decimated. In FIG. 8, the data has been reconstructed using a 3D interpolation algorithm resulting in the image 60. FIG. 8 shows that while the interpolation used performed well when reconstructing the coarse-scale events, the fine-scale event is smeared in space with the amplitude of the fine-scale event being weakened. In FIG. 9, the map view 62 of the model shows the size of the fine-scale event 64 relative to the rest of the area 66 of the map view of the model.
A problem with currently used methods for 5D interpolation can be seen with respect to FIG. 9 in which the map view of the reconstruction of the fine-scale feature 64 is negatively influenced by the large number of surrounding traces that contain no information about the fine-scale feature. However, the surrounding traces support the reconstruction of the coarse-scale events.
Accordingly, it would be desirable to provide methods and systems that avoid the afore-described problems and drawbacks.